Civil Engineering Reference
In-Depth Information
This problem is modeled using rectangular plane stress elements (Chapter 9) and
three meshes used in sequence, as shown (Figure 6.1b-6.1d). Solution conver-
gence is depicted in Figure 6.1e in terms of maximum normal stress in the
x
direction. For this example, the exact solution is taken to be the maximum
bending stress computed using elementary beam theory. The true exact solution
is the plane stress solution from the theory of elasticity. However, the maximum
normal stress is not appreciably changed in the elasticity solution.
The need for convergence during regular mesh refinement is rather clear.
If convergence is not obtained, the engineer using the finite element method
has absolutely no indication whether the results are indicative of a meaningful
approximation to the correct solution. For a general field problem in which the
field variable of interest is expressed on an element basis in the discretized
form
M
(
e
)
(
x
,
y
,
z
)
=
N
i
(
x
,
y
,
z
)
i
(6.1)
i
=
1
where
M
is the number of element degrees of freedom, the interpolation functions
must satisfy two primary conditions to ensure convergence during mesh refine-
ment: the
compatibility
and
completeness
requirements, described as follows.
6.2.1 Compatibility
Along element boundaries, the field variable and its partial derivatives up to one
order less than the highest-order derivative appearing in the integral formula-
tion of the element equations must be continuous.
Given the discretized repre-
sentation of Equation 6.1, it follows that the interpolation functions must meet
this condition, since these functions determine the spatial variation of the field
variable.
Recalling the application of Galerkin's method to the formulation of the
truss element equations, the first derivative of the displacement appears in Equa-
tion 5.34. Therefore, the displacement must be continuous across element bound-
aries, but none of the displacement derivatives is required to be continuous
across such boundaries. Indeed, as observed previously, the truss element is a
constant strain element, so the first derivative is, in general, discontinuous at the
boundaries. Similarly, the beam element formulation, Equation 5.49, includes
the second derivative of displacement, and compatibility requires continuity of
both the displacement and the slope (first derivative) at the element boundaries.
In addition to satisfying the criteria for convergence, the compatibility con-
dition can be given a physical meaning as well. In structural problems, the
requirement of displacement continuity along element boundaries ensures that
no gaps or voids develop in the structure as a result of modeling procedure. Sim-
ilarly the requirement of slope continuity for the beam element ensures that no
“kinks” are developed in the deformed structure. In heat transfer problems, the
compatibility requirement prevents the physically unacceptable possibility of
jump discontinuities in temperature distribution.
